This paper provides a comprehensive error analysis of learning with vector-valued random features (RF). The theory is developed for RF ridge regression in a fully general infinite-dimensional input-output setting, but nonetheless applies to and improves existing finite-dimensional analyses. In contrast to comparable work in the literature, the approach proposed here relies on a direct analysis of the underlying risk functional and completely avoids the explicit RF ridge regression solution formula in terms of random matrices. This removes the need for concentration results in random matrix theory or their generalizations to random operators. The main results established in this paper include strong consistency of vector-valued RF estimators under model misspecification and minimax optimal convergence rates in the well-specified setting. The parameter complexity (number of random features) and sample complexity (number of labeled data) required to achieve such rates are comparable with Monte Carlo intuition and free from logarithmic factors.

We study generalization properties of random features (RF) regression in high dimensions optimized by stochastic gradient descent (SGD) in under-/overparameterized regime. In this work, we derive precise non-asymptotic error bounds of RF regression under both constant and polynomial-decay step-size SGD setting, and observe the double descent phenomenon both theoretically and empirically. Our analysis shows how to cope with multiple randomness sources of initialization, label noise, and data sampling (as well as stochastic gradients) with no closedform solution, and also goes beyond the commonly-used Gaussian/spherical data assumption. Our theoretical results demonstrate that, with SGD training, RF regression still generalizes well for interpolation learning, and is able to characterize the double descent behavior by the unimodality of variance and monotonic decrease of bias. Besides, we also prove that the constant step-size SGD setting incurs no loss in convergence rate when compared to the exact minimum-norm interpolator, as a theoretical justification of using SGD in practice.

Our main contribution in this work is an empirical finding that random General Value Functions (GVFs), i.e., deep action-conditional predictions--random both in what feature of observations they predict as well as in the sequence of actions the predictions are conditioned upon--form good auxiliary tasks for reinforcement learning (RL) problems. In particular, we show that random deep action-conditional predictions when used as auxiliary tasks yield state representations that produce control performance competitive with state-of-the-art hand-crafted auxiliary tasks like value prediction, pixel control, and CURL in both Atari and DeepMind Lab tasks. In another set of experiments we stop the gradients from the RL part of the network to the state representation learning part of the network and show, perhaps surprisingly, that the auxiliary tasks alone are sufficient to learn state representations good enough to outperform an end-to-end trained actor-critic baseline.

C.2 Consolidated CV with random features Alternatively, one can use random features (Rahimi and Recht, 2007) to approximate the kernel matrix. Suppose that we consider shift-invariant kernels that satisfy K(x,y) = K(x y). In this work we use the radial kernel K(x,y) = exp( σ x y 22). The kernel can be approximated by K(x,y) φ(x),φ(y), where an explicit randomized feature mapping φ: IRp IRm is obtained by sampling from a distribution defined by the inverse Fourier transformation.

Randomized features provide a computationally efficient way to approximate kernel machines in machine learning tasks. However, such methods require a user-defined kernel as input. We extend the randomized-feature approach to the task of learning a kernel (via its associated random features). Specifically, we present an efficient optimization problem that learns a kernel in a supervised manner. We prove the consistency of the estimated kernel as well as generalization bounds for the class of estimators induced by the optimized kernel, and we experimentally evaluate our technique on several datasets. Our approach is efficient and highly scalable, and we attain competitive results with a fraction of the training cost of other techniques.

Tabular foundation models (TFMs) such as TabPFN (Tabular Prior-Data Fitted Network) are designed to generalize across heterogeneous tabular datasets through in-context learning (ICL). They perform prediction in a single forward pass conditioned on labeled examples without dataset-specific parameter updates. This paradigm is particularly attractive in industrial domains (e.g., finance and healthcare) where tabular prediction is pervasive. Retraining a bespoke model for each new table can be costly or infeasible in these settings, while data quality issues such as irrelevant predictors, correlated feature groups, and label noise are common. In this paper, we provide strong empirical evidence that TabPFN is highly robust under these sub-optimal conditions. We study TabPFN and its attention mechanisms for binary classification problems with controlled synthetic perturbations that vary: (i) dataset width by injecting random uncorrelated features and by introducing nonlinearly correlated features, (ii) dataset size by increasing the number of training rows, and (iii) label quality by increasing the fraction of mislabeled targets. Beyond predictive performance, we analyze internal signals including attention concentration and attention-based feature ranking metrics. Across these parametric tests, TabPFN is remarkably resilient: ROC-AUC remains high, attention stays structured and sharp, and informative features are highly ranked by attention-based metrics. Qualitative visualizations with attention heatmaps, feature-token embeddings, and SHAP plots further support a consistent pattern across layers in which TabPFN increasingly concentrates on useful features while separating their signals from noise. Together, these findings suggest that TabPFN is a robust TFM capable of maintaining both predictive performance and coherent internal behavior under various scenarios of data imperfections.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/